Subgroups — #20272596

Order 6784.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133128	C₂	1	2	yes	maximal
2	C₅₃×#382	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88322	—	2	2	no	—
4	#13133128	—	2	2	no	—
5	#13133128	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	#95×C₅₃	—	2	2	no	—
9	C₂²×C₄×C₅₃	—	2	2	no	—
10	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	D₄	1	2	yes	—
12	C₄⋊C₄×C₅₃	—	4	2	no	—
13	C₄²×C₅₃	C₂×C₄	1	2	yes	—
14	C₄²×C₅₃	D₄	1	1	yes	characteristic
15	C₄²×C₅₃	Q₈	1	1	yes	characteristic
16	C₂×#88322	—	2	8	no	—
17	C₂×#88322	—	2	4	no	—
18	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₅₃	—	2	1	no	—
20	C₂×C₄×C₅₃	—	2	1	no	—
21	C₂×C₄×C₅₃	—	2	4	no	—
22	C₂×C₄×C₅₃	—	4	2	no	—
23	C₂×C₄×C₅₃	—	4	4	no	—
24	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
26	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
28	#88322	—	2	8	no	—
29	C₂²×C₅₃	#95	1	1	yes	characteristic
30	C₂²×C₅₃	#100	1	1	yes	characteristic
31	C₂²×C₅₃	#101	1	1	yes	characteristic
32	C₂²×C₅₃	#104	1	4	yes	—
33	C₄×C₅₃	—	2	2	no	—
34	C₄×C₅₃	—	2	2	no	—
35	C₄×C₅₃	—	2	1	no	—
36	C₄×C₅₃	—	2	1	no	—
37	C₄×C₅₃	—	8	2	no	—
38	#1295	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	#329	1	2	yes	—
40	C₂×C₅₃	#330	1	2	yes	—
41	C₂×C₅₃	#339	1	2	yes	—
42	C₂×C₅₃	#343	1	1	yes	characteristic
43	C₂×#97	—	53	2	no	—
44	#382	D₅₃	1	1	yes	characteristic
45	C₅₃	#1295	1	1	yes	characteristic, complemented, Sylow
46	C₂²×C₈	—	106	2	no	—
47	C₂×C₄⋊C₄	Dic₅₃	1	2	yes	—
48	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
49	#97	—	53	4	no	—
50	#97	—	106	2	no	—
51	#95	—	2	2	no	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
54	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
55	C₂×C₈	—	106	8	no	—
56	C₂×C₈	—	106	4	no	—
57	C₄⋊C₄	—	4	2	no	—
58	C₄²	Dic₁₀₆	1	1	yes	characteristic
59	C₄²	C₄×D₅₃	1	2	yes	—
60	C₄²	D₂₁₂	1	1	yes	characteristic
61	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	4	4	no	—
67	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
68	C₂×C₄	#11719919	1	2	yes	—
69	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
70	C₂×C₄	#11719921	1	2	yes	—
71	C₈	—	106	8	no	—
72	C₂²	#13133124	1	1	yes	characteristic
73	C₂²	#13133134	1	1	yes	characteristic
74	C₂²	#13133136	1	1	yes	characteristic
75	C₂²	#13133144	1	4	yes	—
76	C₄	—	2	2	no	—
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	8	2	no	—
81	C₂	#16810363	1	2	yes	—
82	C₂	#16810365	1	2	yes	—
83	C₂	#16810382	1	2	yes	—
84	C₂	#16810392	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₅₃×#382	C₂
socle	C₂³×C₅₃	C₂²⋊C₄
hypercenter	#382	D₅₃
hypocenter	C₅₃	#1295
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
5. #13133128	C₄	semidirect
45. C₅₃	#1295	semidirect